England bowler Stuart Broad took 5-73 for Nottinghamshire but centuries from George Bartlett and Tom Abell put Somerset in control on day two.
Resuming on 74-3, Abell made 101 and Bartlett scored a career-best 133 as they added 223 for the fourth wicket.
Lewis Gregory’s run-a-ball 50 helped the visitors to maximum batting points, but Broad took the last three wickets to dismiss Somerset for 403.
Notts closed on 25-2 in their second innings, still trailing by 115 runs.Relive Friday’s County Championship action
The hosts faced a testing nine overs at the end of the day and lost Ben Duckett for four and Chris Nash for a 10-ball duck, with Jack Brooks taking 2-8 in four overs.
Earlier, Bartlett and Abell, who had come together on Thursday evening with Somerset in some trouble on 36-3, frustrated Notts for all of the morning session and much of the afternoon.
Their partnership was finally ended when Abell was late in driving Luke Wood and edged behind to Tom Moores.
Bartlett, 21, spent five hours at the crease as he passed his previous highest score of 110, but eventually left a delivery from Luke Fletcher (3-66) alone and saw his off-stump flattened.
Gregory, who took 6-68 with the ball on day one, then smashed three sixes and four fours – with two successive maximums off Broad – as he made a quickfire half-century.
Broad took three wickets in seven balls to finish off the Somerset innings, but the damage had already been done and the visitors are well placed to make it two wins from two to start the season.
Notts all-rounder Samit Patel missed the first two hours of play following the birth of his baby daughter.N
“It was frustrating not to get a wicket but we bowled fine in that session. The ball didn’t do a lot and I thought Tom Abell and George Bartlett played really well.
“I thought the hour after lunch we didn’t get it right, if I’m honest. With the second new ball we picked up and we saw an outstanding spell of bowling by an international class bowler in Stuart Broad in that session.”S
“I’m really pleased to get there with my second hundred. I was delighted to get that one last year but I was really focused on getting this second one to prove it wasn’t a one-time thing.
“I’m chuffed to bits to get over the line, the partnership with Tom (Abell) was crucial to get us into get into a strong position, so I’m really pleased.”